Short Answer 
The problem involves a circle with radii AB, AE, and AD, and secants BE and ED represented by the expressions 3x Р24 and x + 10, respectively. By proving congruence of triangles ŒiEAB and ŒiEAD, it is determined that EB equals ED, leading to the solution for x as 17, resulting in a length of 27 units for segment BE.
Step 1: Understanding the Circle and Its Segments
In the given circle (circle A), we have three important line segments: AB, AE, and AD, which are all radii of the circle. Meanwhile, the segments BE and ED are classified as secants. Here are their lengths:
- Length of BE is given as 3x – 24.
- Length of ED is given as x + 10.
Step 2: Using Triangle Congruence to Relate Segments
We will establish a relationship between the segments by proving congruence in two triangles, ŒiEAB and ŒiEAD. Since AB and AD are equal (both are radii), we also note that EA is a common side. This allows us to assert:
- Both triangles share a common side EA.
- The angles ‚a†BAE and ‚a†DAE are equal.
- Using the SAS postulate, the triangles are congruent: ŒiEAB ‚aO ŒiEAD.
Step 3: Solve for the Length of BE
From the congruence of the triangles, we conclude that EB is equal to ED, leading us to the equation 3x – 24 = x + 10. Here’s how to solve for x:
- Add 24 to both sides to get: 3x = x + 34.
- Subtract x from both sides, yielding: 2x = 34.
- Divide both sides by 2 to find: x = 17.
- Finally, substitute back into the equation for BE: BE = 3(17) – 24 = 27 units.